# Definition of integration with respect to a vector

I'm trying to understand the definition of the integration with respect to a vector and its relation with Lebesgue integration.

In single variable calculus with $$x$$ is a number the integral $$\int f(x) dx$$ is easily to understand.

However, I have come across many situations where I met the notation $$\int f(x) dx$$, but now $$x = (x_1, ...,x_N)$$ which is a vector of dimension N. In other words, the part $$d$$ (i.e. "with respect to") in this case is "with respect to a vector".

I also have some knowledge of Lebesgue integral to interpret the notation $$\int f d\mu$$ as "integrate f with respect to the measure $$\mu$$". In this case, $$\mu$$ is the measure.

1. What is the rigorous definition of the notation $$\int f(x) dx$$ where $$x$$ is a vector of N dimension and $$f$$ is a real valued function ? What does it mean by saying "with respect to a vector" in this case ?

2. If I want to interpret the above notation $$f(x) dx$$ with respect to Lebesgue integral, what is the measure I shoud use ?

Thank you so much for your help!

For a function $$f:\Bbb{R}^n\to\Bbb{C}$$, the notation $$\int f(x)\,dx$$ means (if nothing else is specified) $$\int_{\Bbb{R}^n}f\,d\lambda_n\equiv \int_{\Bbb{R}^n}f(x)\,d\lambda_n(x)$$, where $$\lambda_n$$ is the Lebesgue measure on $$\Bbb{R}^n$$; so of course for this to make sense one should assume that the function is integrable: $$f\in L^1(\Bbb{R}^n, \mathcal{L}_{\Bbb{R}^n},\lambda_n)\equiv L^1(\Bbb{R}^n)$$. It is also common to omit the subscript $$n$$ in $$\lambda_n$$, because context usually tells us the intended dimension.
We write $$\int f(x)\,dx$$ or $$\int_{\Bbb{R}^n}f(x)\,dx$$ simply for convenience, because on $$\Bbb{R}^n$$, if nothing else is suggested one always uses Lebesgue measure. Sometimes, you might see this written as $$\int_{\Bbb{R}^n}f(x)\,d^nx$$, where the $$d^nx$$ is used to stand for the integration with respect to $$n$$-dimensional Lebesgue measure. For concrete calculations such as in 2 or 3 dimensions, you may see the notations \begin{align} \int_{\Bbb{R}^3}f(x,y,z)\,d\lambda(x,y,z)\quad\text{or}\quad\int_{\Bbb{R}^3}f(x,y,z)\,d(x,y,z) \quad \text{or}\quad\int_{\Bbb{R}^3}f(x,y,z)\,dx\,dy\,dz \end{align} The first two aren't as common, I've seen the third a lot (sometimes the domain of integration is also left implicit). As a super concrete example, let $$f:\Bbb{R}^2\to\Bbb{R}$$ be $$f(x,y)=e^{-(x^2+y^2)}$$. Rather than writing $$\int_{\Bbb{R}^2}f\,d\lambda_2$$, one typically just writes $$\int_{\Bbb{R}^2}e^{-(x^2+y^2)}\,dx\,dy$$ or $$\int_{\Bbb{R}^2}e^{-(x^2+y^2)}\,d(x,y)$$.